Subgroups — C₂²×#66010 #13006225

Order 1280.

14462 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#713	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×D₄×D₅	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₅×#124	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×#4092	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×#4084	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#66010	C₂	1	48	yes	maximal, direct product factor
7	C₂²×#4029	C₂	1	2	yes	maximal, complemented
8	C₂²×#4030	C₂	1	2	yes	maximal, complemented
9	C₂²×#4028	C₂	1	2	yes	maximal, complemented
10	C₂²×D₅×C₂²⋊C₄	C₂	1	2	yes	maximal, complemented
11	C₂²×C₄×D₂₀	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₄×D₂₀	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂⁵×D₅	C₂²	1	2	yes	—
14	C₂³×C₅×Q₈	C₂²	1	1	yes	characteristic, complemented
15	C₂³×C₅×D₄	C₂²	1	1	yes	characteristic
16	C₂³×C₂²⋊D₅	C₂²	1	2	yes	—
17	C₂³×C₂²⋊D₅	C₂²	1	2	yes	complemented
18	C₂²×#713	—	2	16	no	—
19	C₂²×#713	C₂²	1	24	yes	complemented
20	C₂²×D₄×D₅	—	2	16	no	—
21	C₂²×D₄×D₅	C₂²	1	24	yes	complemented
22	C₂³×D₂₀	C₂²	1	1	yes	characteristic
23	C₂³×D₂₀	C₂²	1	1	yes	characteristic
24	C₂³×D₂₀	C₂²	1	2	yes	complemented
25	C₂³×C₄×D₅	C₂²	1	1	yes	characteristic
26	C₂³×C₄×D₅	C₂²	1	1	yes	characteristic, complemented
27	C₂³×C₄×D₅	C₂²	1	2	yes	complemented
28	C₂³×C₄×D₅	C₂²	1	2	yes	complemented
29	C₂×C₅×#124	C₂²	1	24	yes	complemented
30	C₂²×C₅×C₂²⋊C₄	C₂²	1	2	yes	complemented
31	C₂²×C₅×C₂²⋊C₄	C₂²	1	2	yes	complemented
32	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic, complemented
33	C₂²×C₂²⋊Dic₅	C₂²	1	2	yes	complemented
34	C₂×#4092	C₂²	1	24	yes	complemented
35	C₂×#4084	C₂²	1	24	yes	complemented
36	C₂²×#685	C₂²	1	2	yes	—
37	C₂²×#685	C₂²	1	1	yes	characteristic
38	C₂²×#685	C₂²	1	2	yes	complemented
39	C₂²×#685	C₂²	1	1	yes	characteristic, complemented
40	C₂²×C₄⋊Dic₅	C₂²	1	1	yes	characteristic, complemented
41	C₂²×C₄⋊Dic₅	C₂²	1	1	yes	characteristic, complemented
42	C₂²×#683	C₂²	1	2	yes	complemented
43	#66010	C₂²	1	256	yes	direct product factor
44	C₂×#4030	C₂²	1	48	yes	complemented
45	C₂×#4029	C₂²	1	48	yes	complemented
46	C₂×#4028	C₂²	1	48	yes	complemented
47	C₂×D₅×C₂²⋊C₄	C₂²	1	48	yes	complemented
48	C₂×C₄×D₂₀	C₂²	1	24	yes	complemented
49	C₂×C₄×D₂₀	C₂²	1	24	yes	complemented
50	C₂²×#546	—	5	1	no	maximal, Sylow
51	C₂⁵×C₅	C₂³	1	2	yes	—
52	C₂⁴×D₅	—	2	16	no	—
53	C₂⁴×D₅	—	2	16	no	—
54	C₂⁴×D₅	—	4	8	no	—
55	C₂⁴×D₅	C₂³	1	1	yes	characteristic
56	C₂⁴×D₅	C₂³	1	24	yes	—
57	C₂⁴×D₅	C₂³	1	2	yes	—
58	C₂⁴×D₅	C₂³	1	1	yes	characteristic
59	C₂²×C₅×Q₈	—	2	8	no	—
60	C₂²×C₅×Q₈	C₂³	1	12	yes	complemented
61	C₂²×C₅×D₄	—	2	8	no	—
62	C₂²×C₅×D₄	C₂³	1	12	yes	—
63	C₂³×C₄×C₅	C₂³	1	2	yes	—
64	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
65	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
66	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
67	C₂²×C₂²⋊D₅	—	2	16	no	—
68	C₂²×C₂²⋊D₅	—	4	8	no	—
69	C₂²×C₂²⋊D₅	C₂³	1	24	yes	—
70	C₂²×C₂²⋊D₅	C₂³	1	24	yes	complemented
71	C₂³×Dic₅	C₂³	1	1	yes	characteristic
72	C₂³×Dic₅	C₂³	1	2	yes	—
73	C₂³×Dic₅	C₂³	1	1	yes	characteristic
74	C₂×#713	—	2	192	no	—
75	C₂×#713	C₂³	1	64	yes	complemented
76	C₂×D₄×D₅	—	2	192	no	—
77	C₂×D₄×D₅	C₂³	1	64	yes	complemented
78	C₂²×D₂₀	—	2	16	no	—
79	C₂²×D₂₀	—	2	8	no	—
80	C₂²×D₂₀	D₄	1	16	yes	complemented
81	C₂²×D₂₀	C₂³	1	12	yes	—
82	C₂²×D₂₀	C₂³	1	12	yes	—
83	C₂²×D₂₀	C₂³	1	24	yes	complemented
84	C₂²×C₄×D₅	—	2	16	no	—
85	C₂²×C₄×D₅	—	2	8	no	—
86	C₂²×C₄×D₅	—	2	8	no	—
87	C₂²×C₄×D₅	—	4	8	no	—
88	C₂²×C₄×D₅	C₂³	1	12	yes	—
89	C₂²×C₄×D₅	C₂³	1	12	yes	complemented
90	C₂²×C₄×D₅	C₂³	1	24	yes	complemented
91	C₂²×C₄×D₅	C₂³	1	24	yes	complemented
92	C₅×#124	C₂³	1	64	yes	complemented
93	C₂×C₅×C₂²⋊C₄	C₂³	1	24	yes	complemented
94	C₂×C₅×C₂²⋊C₄	C₂³	1	24	yes	complemented
95	C₂×C₄²×C₅	C₂³	1	12	yes	complemented
96	C₂×C₂²⋊Dic₅	C₂³	1	24	yes	complemented
97	#4092	C₂³	1	64	yes	complemented
98	#4084	C₂³	1	64	yes	complemented
99	C₂×#685	C₂³	1	12	yes	—
100	C₂×#685	C₂³	1	24	yes	—
101	C₂×#685	C₂³	1	24	yes	complemented
102	C₂×#685	C₂³	1	12	yes	complemented
103	C₂×C₄⋊Dic₅	C₂³	1	12	yes	complemented
104	C₂×C₄⋊Dic₅	C₂³	1	12	yes	complemented
105	C₂×#683	C₂³	1	24	yes	complemented
106	#4030	C₂³	1	128	yes	complemented
107	#4029	C₂³	1	128	yes	complemented
108	#4028	C₂³	1	128	yes	complemented
109	D₅×C₂²⋊C₄	C₂³	1	128	yes	complemented
110	C₄×D₂₀	C₂³	1	64	yes	complemented
111	C₄×D₂₀	C₂³	1	64	yes	complemented
112	C₂³×#41	—	5	1	no	—
113	C₂⁴×D₄	—	5	1	no	—
114	C₂×#546	—	5	48	no	—
115	C₂²×#124	D₅	1	1	yes	characteristic, complemented
116	C₂²×#123	—	5	2	no	—
117	C₂²×C₂²⋊Q₈	—	5	1	no	—
118	C₂²×#121	—	5	1	no	—
119	C₂²×#121	—	5	2	no	—
120	C₂²×#120	—	5	2	no	—
121	C₂²×C₄×D₄	—	5	1	no	—
122	C₂²×C₄×D₄	—	5	1	no	—
123	C₂³×C₂²⋊C₄	—	5	2	no	—
124	C₂⁴×C₅	—	2	8	no	—
125	C₂⁴×C₅	—	2	8	no	—
126	C₂⁴×C₅	—	2	8	no	—
127	C₂⁴×C₅	C₂⁴	1	12	yes	—
128	C₂⁴×C₅	C₂⁴	1	1	yes	characteristic
129	C₂³×D₅	—	2	4	no	—
130	C₂³×D₅	—	2	8	no	—
131	C₂³×D₅	—	2	8	no	—
132	C₂³×D₅	—	2	4	no	—
133	C₂³×D₅	—	2	4	no	—
134	C₂³×D₅	—	2	4	no	—
135	C₂³×D₅	—	2	4	no	—
136	C₂³×D₅	—	2	96	no	—
137	C₂³×D₅	—	2	96	no	—
138	C₂³×D₅	—	4	48	no	—
139	C₂³×D₅	—	4	128	no	—
140	C₂³×D₅	C₂×D₄	1	16	yes	—
141	C₂³×D₅	#41	1	8	yes	complemented
142	C₂³×D₅	C₂⁴	1	6	yes	—
143	C₂³×D₅	C₂⁴	1	32	yes	—
144	C₂³×D₅	C₂⁴	1	12	yes	—
145	C₂³×D₅	C₂⁴	1	6	yes	—
146	C₂×C₅×Q₈	—	2	48	no	—
147	C₂×C₅×Q₈	C₂⁴	1	16	yes	complemented
148	C₂×C₅×D₄	—	2	48	no	—
149	C₂×C₅×D₄	C₂⁴	1	16	yes	—
150	C₂²×C₄×C₅	—	2	8	no	—
151	C₂²×C₄×C₅	—	2	4	no	—
152	C₂²×C₄×C₅	—	2	4	no	—
153	C₂²×C₄×C₅	C₂×D₄	1	8	yes	—
154	C₂²×C₄×C₅	C₂⁴	1	6	yes	—
155	C₂²×C₄×C₅	C₂⁴	1	6	yes	—
156	C₂²×C₄×C₅	C₂⁴	1	12	yes	—
157	C₂²×C₄×C₅	C₂⁴	1	6	yes	—
158	C₂×C₂²⋊D₅	—	2	96	no	—
159	C₂×C₂²⋊D₅	—	4	48	no	—
160	C₂×C₂²⋊D₅	C₂⁴	1	32	yes	—
161	C₂×C₂²⋊D₅	C₂⁴	1	32	yes	complemented
162	C₂²×Dic₅	—	2	4	no	—
163	C₂²×Dic₅	—	2	4	no	—
164	C₂²×Dic₅	—	2	8	no	—
165	C₂²×Dic₅	C₂⁴	1	6	yes	—
166	C₂²×Dic₅	C₂⁴	1	12	yes	—
167	C₂²×Dic₅	C₂⁴	1	6	yes	—
168	#713	—	2	256	no	—
169	D₄×D₅	—	2	256	no	—
170	C₂×D₂₀	—	2	96	no	—
171	C₂×D₂₀	—	2	48	no	—
172	C₂×D₂₀	C₂×D₄	1	96	yes	complemented
173	C₂×D₂₀	C₂⁴	1	16	yes	—
174	C₂×D₂₀	C₂⁴	1	16	yes	—
175	C₂×D₂₀	C₂⁴	1	32	yes	complemented
176	C₂×C₄×D₅	—	2	96	no	—
177	C₂×C₄×D₅	—	2	48	no	—
178	C₂×C₄×D₅	—	2	48	no	—
179	C₂×C₄×D₅	—	4	48	no	—
180	C₂×C₄×D₅	C₂⁴	1	16	yes	—
181	C₂×C₄×D₅	C₂⁴	1	16	yes	complemented
182	C₂×C₄×D₅	C₂⁴	1	32	yes	complemented
183	C₂×C₄×D₅	C₂⁴	1	32	yes	complemented
184	C₅×C₂²⋊C₄	C₂⁴	1	32	yes	complemented
185	C₅×C₂²⋊C₄	C₂⁴	1	32	yes	complemented
186	C₄²×C₅	C₂⁴	1	16	yes	complemented
187	C₂²⋊Dic₅	C₂⁴	1	32	yes	complemented
188	#685	C₂⁴	1	16	yes	—
189	#685	C₂⁴	1	32	yes	—
190	#685	C₂⁴	1	32	yes	complemented
191	#685	C₂⁴	1	16	yes	complemented
192	C₄⋊Dic₅	C₂⁴	1	16	yes	complemented
193	C₄⋊Dic₅	C₂⁴	1	16	yes	complemented
194	#683	C₂⁴	1	32	yes	complemented
195	C₂⁶	—	5	2	no	—
196	C₂²×#41	—	5	24	no	—
197	C₂²×#41	—	10	16	no	—
198	C₂³×Q₈	C₂×D₅	1	1	yes	characteristic, complemented
199	C₂³×D₄	—	5	2	no	—
200	C₂³×D₄	—	5	2	no	—
201	C₂³×D₄	—	5	24	no	—
202	C₂³×D₄	—	5	1	no	—
203	C₂³×D₄	—	5	2	no	—
204	C₂³×D₄	—	5	1	no	—
205	C₂³×D₄	—	10	16	no	—
206	C₂³×D₄	C₂×D₅	1	1	yes	characteristic
207	C₂⁴×C₄	—	5	1	no	—
208	C₂⁴×C₄	—	5	2	no	—
209	C₂⁴×C₄	—	5	1	no	—
210	C₂⁴×C₄	—	5	2	no	—
211	#546	—	5	256	no	—
212	C₂×#124	C₂×D₅	1	24	yes	complemented
213	C₂×#123	—	5	48	no	—
214	C₂×C₂²⋊Q₈	—	5	24	no	—
215	C₂×#121	—	5	24	no	—
216	C₂×#121	—	5	48	no	—
217	C₂×#120	—	5	48	no	—
218	C₂×C₄×D₄	—	5	24	no	—
219	C₂×C₄×D₄	—	5	24	no	—
220	C₂²×C₄⋊C₄	—	5	2	no	—
221	C₂²×C₄⋊C₄	—	5	1	no	—
222	C₂²×C₄⋊C₄	—	5	1	no	—
223	C₂²×C₂²⋊C₄	—	5	2	no	—
224	C₂²×C₂²⋊C₄	—	5	1	no	—
225	C₂²×C₂²⋊C₄	—	5	2	no	—
226	C₂²×C₂²⋊C₄	—	5	2	no	—
227	C₂²×C₂²⋊C₄	—	5	1	no	—
228	C₂²×C₂²⋊C₄	—	5	48	no	—
229	C₂²×C₂²⋊C₄	C₂×D₅	1	2	yes	complemented
230	C₂²×C₂²⋊C₄	C₂×D₅	1	2	yes	complemented
231	C₂²×C₄²	C₂×D₅	1	1	yes	characteristic, complemented
232	C₂³×C₅	—	2	24	no	—
233	C₂³×C₅	—	2	24	no	—
234	C₂³×C₅	—	2	24	no	—
235	C₂³×C₅	—	4	32	no	—
236	C₂³×C₅	C₂²×D₄	1	4	yes	—
237	C₂³×C₅	C₂×#41	1	4	yes	—
238	C₂³×C₅	H(5,𝔽₂)	1	4	yes	—
239	C₂³×C₅	C₂⁵	1	8	yes	—
240	C₂³×C₅	C₂⁵	1	3	yes	—
241	C₂²×D₅	—	2	32	no	—
242	C₂²×D₅	—	2	64	no	—
243	C₂²×D₅	—	2	12	no	—
244	C₂²×D₅	—	2	24	no	—
245	C₂²×D₅	—	2	24	no	—
246	C₂²×D₅	—	2	12	no	—
247	C₂²×D₅	—	2	12	no	—
248	C₂²×D₅	—	2	12	no	—
249	C₂²×D₅	—	2	12	no	—
250	C₂²×D₅	—	2	64	no	—
251	C₂²×D₅	—	2	64	no	—
252	C₂²×D₅	—	4	32	no	—
253	C₂²×D₅	—	4	384	no	—
254	C₂²×D₅	—	4	16	no	—
255	C₂²×D₅	C₂²×D₄	1	48	yes	—
256	C₂²×D₅	C₂×#41	1	24	yes	complemented
257	C₂²×D₅	C₂⁵	1	4	yes	—
258	C₂²×D₅	C₂⁵	1	8	yes	—
259	C₂²×D₅	C₂⁵	1	4	yes	—
260	C₅×Q₈	—	2	32	no	—
261	C₅×D₄	—	2	32	no	—
262	C₂×C₄×C₅	—	2	24	no	—
263	C₂×C₄×C₅	—	2	12	no	—
264	C₂×C₄×C₅	—	2	12	no	—
265	C₂×C₄×C₅	C₂²×D₄	1	24	yes	—
266	C₂×C₄×C₅	C₂⁵	1	8	yes	—
267	C₂×C₄×C₅	C₂⁵	1	4	yes	—
268	C₂×C₄×C₅	C₂⁵	1	4	yes	—
269	C₂×C₄×C₅	C₂⁵	1	4	yes	—
270	C₂²⋊D₅	—	2	64	no	—
271	C₂²⋊D₅	—	4	32	no	—
272	C₂×Dic₅	—	2	12	no	—
273	C₂×Dic₅	—	2	12	no	—
274	C₂×Dic₅	—	2	24	no	—
275	C₂×Dic₅	C₂⁵	1	4	yes	—
276	C₂×Dic₅	C₂⁵	1	8	yes	—
277	C₂×Dic₅	C₂⁵	1	4	yes	—
278	D₂₀	—	2	64	no	—
279	D₂₀	—	2	32	no	—
280	D₂₀	C₂²×D₄	1	64	yes	complemented
281	C₄×D₅	—	2	64	no	—
282	C₄×D₅	—	2	32	no	—
283	C₄×D₅	—	2	32	no	—
284	C₄×D₅	—	4	32	no	—
285	C₂⁵	—	5	1	no	—
286	C₂⁵	—	5	2	no	—
287	C₂⁵	—	5	1	no	—
288	C₂⁵	—	5	24	no	—
289	C₂⁵	—	10	16	no	—
290	C₂⁵	—	10	16	no	—
291	C₂⁵	—	20	8	no	—
292	C₂⁵	C₂²×D₅	1	2	yes	—
293	C₂×#41	—	5	64	no	—
294	C₂×#41	—	10	192	no	—
295	C₂²×Q₈	—	2	8	no	—
296	C₂²×Q₈	C₂²×D₅	1	12	yes	complemented
297	C₂²×D₄	—	2	8	no	—
298	C₂²×D₄	—	5	24	no	—
299	C₂²×D₄	—	5	24	no	—
300	C₂²×D₄	—	5	64	no	—
301	C₂²×D₄	—	5	16	no	—
302	C₂²×D₄	—	5	12	no	—
303	C₂²×D₄	—	5	12	no	—
304	C₂²×D₄	—	5	24	no	—
305	C₂²×D₄	—	10	8	no	—
306	C₂²×D₄	—	10	16	no	—
307	C₂²×D₄	—	10	192	no	—
308	C₂²×D₄	—	10	16	no	—
309	C₂²×D₄	—	20	8	no	—
310	C₂²×D₄	C₂²×D₅	1	12	yes	—
311	C₂³×C₄	—	5	2	no	—
312	C₂³×C₄	—	5	1	no	—
313	C₂³×C₄	—	5	1	no	—
314	C₂³×C₄	—	5	24	no	—
315	C₂³×C₄	—	5	12	no	—
316	C₂³×C₄	—	5	12	no	—
317	C₂³×C₄	—	5	24	no	—
318	C₂³×C₄	—	10	16	no	—
319	C₂³×C₄	—	10	8	no	—
320	C₂³×C₄	—	10	8	no	—
321	C₂³×C₄	—	20	8	no	—
322	C₂³×C₄	C₂²×D₅	1	1	yes	characteristic
323	C₂³×C₄	C₂²×D₅	1	1	yes	characteristic
324	C₂³×C₄	C₂²×D₅	1	2	yes	—
325	C₂³×C₄	C₂²×D₅	1	1	yes	characteristic
326	#124	C₂²×D₅	1	64	yes	complemented
327	#123	—	5	128	no	—
328	C₂²⋊Q₈	—	5	64	no	—
329	#121	—	5	64	no	—
330	#121	—	5	128	no	—
331	#120	—	5	128	no	—
332	C₄×D₄	—	5	64	no	—
333	C₄×D₄	—	5	64	no	—
334	C₂×C₄⋊C₄	—	5	12	no	—
335	C₂×C₄⋊C₄	—	5	12	no	—
336	C₂×C₄⋊C₄	—	5	24	no	—
337	C₂×C₂²⋊C₄	—	5	24	no	—
338	C₂×C₂²⋊C₄	—	5	12	no	—
339	C₂×C₂²⋊C₄	—	5	24	no	—
340	C₂×C₂²⋊C₄	—	5	24	no	—
341	C₂×C₂²⋊C₄	—	5	12	no	—
342	C₂×C₂²⋊C₄	—	5	128	no	—
343	C₂×C₂²⋊C₄	C₂²×D₅	1	24	yes	complemented
344	C₂×C₂²⋊C₄	C₂²×D₅	1	24	yes	complemented
345	C₂×C₄²	C₂²×D₅	1	12	yes	complemented
346	C₂²×C₅	—	2	8	no	—
347	C₂²×C₅	—	2	8	no	—
348	C₂²×C₅	—	2	8	no	—
349	C₂²×C₅	—	4	48	no	—
350	C₂²×C₅	#546	1	16	yes	complemented
351	C₂²×C₅	C₂³×D₄	1	6	yes	—
352	C₂²×C₅	C₂²×#41	1	6	yes	—
353	C₂²×C₅	C₂×H(5,𝔽₂)	1	6	yes	—
354	C₂²×C₅	C₂⁶	1	1	yes	characteristic
355	C₂×D₅	—	2	4	no	—
356	C₂×D₅	—	2	8	no	—
357	C₂×D₅	—	2	8	no	—
358	C₂×D₅	—	2	4	no	—
359	C₂×D₅	—	2	48	no	—
360	C₂×D₅	—	2	96	no	—
361	C₂×D₅	—	2	4	no	—
362	C₂×D₅	—	2	4	no	—
363	C₂×D₅	—	2	4	no	—
364	C₂×D₅	—	4	128	no	—
365	C₂×D₅	—	4	24	no	—
366	C₂×D₅	C₂³×D₄	1	16	yes	—
367	C₂×D₅	C₂²×#41	1	8	yes	complemented
368	C₄×C₅	—	2	8	no	—
369	C₄×C₅	—	2	4	no	—
370	C₄×C₅	—	2	4	no	—
371	C₄×C₅	C₂³×D₄	1	8	yes	—
372	Dic₅	—	2	4	no	—
373	Dic₅	—	2	4	no	—
374	Dic₅	—	2	8	no	—
375	C₂⁴	—	2	8	no	—
376	C₂⁴	—	2	8	no	—
377	C₂⁴	—	2	8	no	—
378	C₂⁴	—	5	16	no	—
379	C₂⁴	—	5	8	no	—
380	C₂⁴	—	5	6	no	—
381	C₂⁴	—	5	12	no	—
382	C₂⁴	—	5	6	no	—
383	C₂⁴	—	5	32	no	—
384	C₂⁴	—	10	96	no	—
385	C₂⁴	—	10	96	no	—
386	C₂⁴	—	10	4	no	—
387	C₂⁴	—	10	8	no	—
388	C₂⁴	—	10	8	no	—
389	C₂⁴	—	10	4	no	—
390	C₂⁴	—	10	4	no	—
391	C₂⁴	—	10	4	no	—
392	C₂⁴	—	10	4	no	—
393	C₂⁴	—	20	48	no	—
394	C₂⁴	—	20	128	no	—
395	C₂⁴	C₂³×D₅	1	12	yes	—
396	C₂⁴	C₂³×D₅	1	1	yes	characteristic
397	#41	—	10	256	no	—
398	C₂×Q₈	—	2	48	no	—
399	C₂×Q₈	C₂³×D₅	1	16	yes	complemented
400	C₂×D₄	—	2	48	no	—
401	C₂×D₄	—	5	32	no	—
402	C₂×D₄	—	5	32	no	—
403	C₂×D₄	—	5	96	no	—
404	C₂×D₄	—	5	16	no	—
405	C₂×D₄	—	5	16	no	—
406	C₂×D₄	—	5	32	no	—
407	C₂×D₄	—	10	96	no	—
408	C₂×D₄	—	10	48	no	—
409	C₂×D₄	—	10	256	no	—
410	C₂×D₄	—	10	96	no	—
411	C₂×D₄	—	20	48	no	—
412	C₂×D₄	C₂³×D₅	1	16	yes	—
413	C₂²×C₄	—	2	8	no	—
414	C₂²×C₄	—	2	4	no	—
415	C₂²×C₄	—	2	4	no	—
416	C₂²×C₄	—	5	12	no	—
417	C₂²×C₄	—	5	6	no	—
418	C₂²×C₄	—	5	6	no	—
419	C₂²×C₄	—	5	32	no	—
420	C₂²×C₄	—	5	16	no	—
421	C₂²×C₄	—	5	16	no	—
422	C₂²×C₄	—	5	32	no	—
423	C₂²×C₄	—	10	96	no	—
424	C₂²×C₄	—	10	4	no	—
425	C₂²×C₄	—	10	4	no	—
426	C₂²×C₄	—	10	8	no	—
427	C₂²×C₄	—	10	48	no	—
428	C₂²×C₄	—	10	48	no	—
429	C₂²×C₄	—	20	48	no	—
430	C₂²×C₄	D₄×D₅	1	8	yes	—
431	C₂²×C₄	C₂³×D₅	1	6	yes	—
432	C₂²×C₄	C₂³×D₅	1	6	yes	—
433	C₂²×C₄	C₂³×D₅	1	12	yes	—
434	C₂²×C₄	C₂³×D₅	1	6	yes	—
435	C₄⋊C₄	—	5	16	no	—
436	C₄⋊C₄	—	5	16	no	—
437	C₄⋊C₄	—	5	32	no	—
438	C₂²⋊C₄	—	5	32	no	—
439	C₂²⋊C₄	—	5	16	no	—
440	C₂²⋊C₄	—	5	32	no	—
441	C₂²⋊C₄	—	5	32	no	—
442	C₂²⋊C₄	—	5	16	no	—
443	C₂²⋊C₄	C₂³×D₅	1	32	yes	complemented
444	C₂²⋊C₄	C₂³×D₅	1	32	yes	complemented
445	C₄²	C₂³×D₅	1	16	yes	complemented
446	C₂×C₅	—	4	8	no	—
447	C₂×C₅	C₂×#546	1	12	yes	complemented
448	C₂×C₅	C₂⁴×D₄	1	1	yes	characteristic
449	C₂×C₅	C₂³×#41	1	1	yes	characteristic
450	C₂×C₅	C₂²×H(5,𝔽₂)	1	1	yes	characteristic
451	D₅	—	2	8	no	—
452	D₅	—	2	16	no	—
453	D₅	—	4	4	no	—
454	C₂³	—	2	24	no	—
455	C₂³	—	2	24	no	—
456	C₂³	—	2	24	no	—
457	C₂³	—	4	32	no	—
458	C₂³	—	5	48	no	—
459	C₂³	—	5	24	no	—
460	C₂³	—	5	4	no	—
461	C₂³	—	5	8	no	—
462	C₂³	—	5	4	no	—
463	C₂³	—	10	64	no	—
464	C₂³	—	10	64	no	—
465	C₂³	—	10	32	no	—
466	C₂³	—	10	64	no	—
467	C₂³	—	10	12	no	—
468	C₂³	—	10	24	no	—
469	C₂³	—	10	24	no	—
470	C₂³	—	10	12	no	—
471	C₂³	—	10	12	no	—
472	C₂³	—	10	12	no	—
473	C₂³	—	10	12	no	—
474	C₂³	—	20	32	no	—
475	C₂³	—	20	384	no	—
476	C₂³	—	20	16	no	—
477	C₂³	C₂×D₄×D₅	1	4	yes	—
478	C₂³	D₅×#41	1	4	yes	—
479	C₂³	#4149	1	4	yes	—
480	C₂³	C₂⁴×D₅	1	8	yes	—
481	C₂³	C₂⁴×D₅	1	3	yes	—
482	Q₈	—	2	32	no	—
483	D₄	—	2	32	no	—
484	D₄	—	5	64	no	—
485	D₄	—	10	64	no	—
486	D₄	—	10	32	no	—
487	D₄	—	10	64	no	—
488	D₄	—	20	32	no	—
489	C₂×C₄	—	2	24	no	—
490	C₂×C₄	—	2	12	no	—
491	C₂×C₄	—	2	12	no	—
492	C₂×C₄	—	5	8	no	—
493	C₂×C₄	—	5	4	no	—
494	C₂×C₄	—	5	4	no	—
495	C₂×C₄	—	10	64	no	—
496	C₂×C₄	—	10	12	no	—
497	C₂×C₄	—	10	12	no	—
498	C₂×C₄	—	10	24	no	—
499	C₂×C₄	—	10	32	no	—
500	C₂×C₄	—	10	32	no	—
501	C₂×C₄	—	20	32	no	—
502	C₂×C₄	C₂×D₄×D₅	1	24	yes	—
503	C₂×C₄	C₂⁴×D₅	1	4	yes	—
504	C₂×C₄	C₂⁴×D₅	1	4	yes	—
505	C₂×C₄	C₂⁴×D₅	1	8	yes	—
506	C₂×C₄	C₂⁴×D₅	1	4	yes	—
507	C₅	C₂²×#546	1	1	yes	characteristic, complemented, Sylow
508	C₂²	—	2	8	no	—
509	C₂²	—	2	8	no	—
510	C₂²	—	2	8	no	—
511	C₂²	—	4	48	no	—
512	C₂²	—	5	16	no	—
513	C₂²	—	5	8	no	—
514	C₂²	—	10	4	no	—
515	C₂²	—	10	8	no	—
516	C₂²	—	10	8	no	—
517	C₂²	—	10	4	no	—
518	C₂²	—	10	48	no	—
519	C₂²	—	10	96	no	—
520	C₂²	—	10	4	no	—
521	C₂²	—	10	4	no	—
522	C₂²	—	10	4	no	—
523	C₂²	—	20	128	no	—
524	C₂²	—	20	24	no	—
525	C₂²	#66010	1	16	yes	direct product factor
526	C₂²	C₂²×D₄×D₅	1	6	yes	—
527	C₂²	C₂×D₅×#41	1	6	yes	—
528	C₂²	C₂×#4149	1	6	yes	—
529	C₂²	C₂⁵×D₅	1	1	yes	characteristic
530	C₄	—	2	8	no	—
531	C₄	—	2	4	no	—
532	C₄	—	2	4	no	—
533	C₄	—	10	4	no	—
534	C₄	—	10	4	no	—
535	C₄	—	10	8	no	—
536	C₄	C₂²×D₄×D₅	1	8	yes	—
537	C₂	—	4	8	no	—
538	C₂	—	10	8	no	—
539	C₂	—	10	16	no	—
540	C₂	—	20	4	no	—
541	C₂	C₂×#66010	1	12	yes	direct product factor
542	C₂	C₂³×D₄×D₅	1	1	yes	characteristic
543	C₂	C₂²×D₅×#41	1	1	yes	characteristic
544	C₂	C₂²×#4149	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂³×D₅
commutator, cosocle	C₂²×C₅	C₂⁶
Frattini	C₂²	C₂⁵×D₅
Fitting	C₂²×C₅×#124	C₂
socle	C₂⁴×C₅	C₂⁴
hypercenter	C₂²×#124	D₅
hypocenter	C₅	C₂²×#546
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂⁴ ◁ C₂²×#124

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂³×#713	C₂	semidirect
2. C₂³×D₄×D₅	C₂	semidirect
3. C₂²×C₅×#124	C₂	semidirect
4. C₂²×#4092	C₂	semidirect
5. C₂²×#4084	C₂	semidirect
6. C₂×#66010	C₂	direct (N × H)
7. C₂²×#4029	C₂	semidirect
8. C₂²×#4030	C₂	semidirect
9. C₂²×#4028	C₂	semidirect
10. C₂²×D₅×C₂²⋊C₄	C₂	semidirect
11. C₂²×C₄×D₂₀	C₂	semidirect
12. C₂²×C₄×D₂₀	C₂	semidirect
14. C₂³×C₅×Q₈	C₂²	semidirect
17. C₂³×C₂²⋊D₅	C₂²	semidirect
19. C₂²×#713	C₂²	semidirect
21. C₂²×D₄×D₅	C₂²	semidirect
24. C₂³×D₂₀	C₂²	semidirect
26. C₂³×C₄×D₅	C₂²	semidirect
27. C₂³×C₄×D₅	C₂²	semidirect
28. C₂³×C₄×D₅	C₂²	semidirect
29. C₂×C₅×#124	C₂²	semidirect
30. C₂²×C₅×C₂²⋊C₄	C₂²	semidirect
31. C₂²×C₅×C₂²⋊C₄	C₂²	semidirect
32. C₂²×C₄²×C₅	C₂²	semidirect
33. C₂²×C₂²⋊Dic₅	C₂²	semidirect
34. C₂×#4092	C₂²	semidirect
35. C₂×#4084	C₂²	semidirect
38. C₂²×#685	C₂²	semidirect
39. C₂²×#685	C₂²	semidirect
40. C₂²×C₄⋊Dic₅	C₂²	semidirect
41. C₂²×C₄⋊Dic₅	C₂²	semidirect
42. C₂²×#683	C₂²	semidirect
43. #66010	C₂²	direct (N × H)
44. C₂×#4030	C₂²	semidirect
45. C₂×#4029	C₂²	semidirect
46. C₂×#4028	C₂²	semidirect
47. C₂×D₅×C₂²⋊C₄	C₂²	semidirect
48. C₂×C₄×D₂₀	C₂²	semidirect
49. C₂×C₄×D₂₀	C₂²	semidirect
60. C₂²×C₅×Q₈	C₂³	semidirect
70. C₂²×C₂²⋊D₅	C₂³	semidirect
75. C₂×#713	C₂³	semidirect
77. C₂×D₄×D₅	C₂³	semidirect
80. C₂²×D₂₀	D₄	semidirect
83. C₂²×D₂₀	C₂³	semidirect
89. C₂²×C₄×D₅	C₂³	semidirect
90. C₂²×C₄×D₅	C₂³	semidirect
91. C₂²×C₄×D₅	C₂³	semidirect
92. C₅×#124	C₂³	semidirect
93. C₂×C₅×C₂²⋊C₄	C₂³	semidirect
94. C₂×C₅×C₂²⋊C₄	C₂³	semidirect
95. C₂×C₄²×C₅	C₂³	semidirect
96. C₂×C₂²⋊Dic₅	C₂³	semidirect
97. #4092	C₂³	semidirect
98. #4084	C₂³	semidirect
101. C₂×#685	C₂³	semidirect
102. C₂×#685	C₂³	semidirect
103. C₂×C₄⋊Dic₅	C₂³	semidirect
104. C₂×C₄⋊Dic₅	C₂³	semidirect
105. C₂×#683	C₂³	semidirect
106. #4030	C₂³	semidirect
107. #4029	C₂³	semidirect
108. #4028	C₂³	semidirect
109. D₅×C₂²⋊C₄	C₂³	semidirect
110. C₄×D₂₀	C₂³	semidirect
111. C₄×D₂₀	C₂³	semidirect
115. C₂²×#124	D₅	semidirect
141. C₂³×D₅	#41	semidirect
147. C₂×C₅×Q₈	C₂⁴	semidirect
161. C₂×C₂²⋊D₅	C₂⁴	semidirect
172. C₂×D₂₀	C₂×D₄	semidirect
175. C₂×D₂₀	C₂⁴	semidirect
181. C₂×C₄×D₅	C₂⁴	semidirect
182. C₂×C₄×D₅	C₂⁴	semidirect
183. C₂×C₄×D₅	C₂⁴	semidirect
184. C₅×C₂²⋊C₄	C₂⁴	semidirect
185. C₅×C₂²⋊C₄	C₂⁴	semidirect
186. C₄²×C₅	C₂⁴	semidirect
187. C₂²⋊Dic₅	C₂⁴	semidirect
190. #685	C₂⁴	semidirect
191. #685	C₂⁴	semidirect
192. C₄⋊Dic₅	C₂⁴	semidirect
193. C₄⋊Dic₅	C₂⁴	semidirect
194. #683	C₂⁴	semidirect
198. C₂³×Q₈	C₂×D₅	semidirect
212. C₂×#124	C₂×D₅	semidirect
229. C₂²×C₂²⋊C₄	C₂×D₅	semidirect
230. C₂²×C₂²⋊C₄	C₂×D₅	semidirect
231. C₂²×C₄²	C₂×D₅	semidirect
256. C₂²×D₅	C₂×#41	semidirect
280. D₂₀	C₂²×D₄	semidirect
296. C₂²×Q₈	C₂²×D₅	semidirect
326. #124	C₂²×D₅	semidirect
343. C₂×C₂²⋊C₄	C₂²×D₅	semidirect
344. C₂×C₂²⋊C₄	C₂²×D₅	semidirect
345. C₂×C₄²	C₂²×D₅	semidirect
350. C₂²×C₅	#546	semidirect
367. C₂×D₅	C₂²×#41	semidirect
399. C₂×Q₈	C₂³×D₅	semidirect
443. C₂²⋊C₄	C₂³×D₅	semidirect
444. C₂²⋊C₄	C₂³×D₅	semidirect
445. C₄²	C₂³×D₅	semidirect
447. C₂×C₅	C₂×#546	semidirect
507. C₅	C₂²×#546	semidirect
525. C₂²	#66010	direct (N × H)
541. C₂	C₂×#66010	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂³×#713	2
C₂³×D₄×D₅	2
C₂²×C₅×#124	2
C₂²×#4092	2
C₂²×#4084	2
C₂×#66010	2
C₂²×#4029	2
C₂²×#4030	2
C₂²×#4028	2
C₂²×D₅×C₂²⋊C₄	2
C₂²×C₄×D₂₀	2
C₂²×#546	5

Subgroups — C22×#66010 #13006225

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#66010 #13006225